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On pg. 85 of The Rise and Development of Theory of Series up to the Early 1820s by Ferraro is a series soln. of

$$ d^2z/z = -x^2dx^2 $$

related to the reputed first appearance of a Riccati-type eqn., addressed by the Bernoulli brothers,

$$dy = y^2dx+x^2dx $$

where $y=-\frac{dz}{zdx}$, an early appearance of the so-called Cole-Hopf transformation.

The series is

$$z= 1-x^4/(3 \cdot 4)+x^8/(3 \cdot 4 \cdot 7 \cdot 8) - x^{12}/(3 \cdot 4 \cdot 7 \cdot 8 \cdot 11 \cdot 12) + .... $$

and the denominators are $d_n \rightarrow (1,12,672,88704...)$.

Looking for connections to enumerative combinatorics, I found the first three terms of the series in the 5th row of the square table rep of OEIS A060638, related to the combinatorics of zonotopes.

Does this relationship to the OEIS entry persist, at least empirically, for higher order terms?

This series is related to two periodic subsets of A214916 by the defining diff eq here and the recursion relation for the OEIS entry:

$$ z = 1 - x^4/12 + x^8/672 - x^{12}/88704 + ... $$

$$= 1 - 2 \; x^4/4! + 60 \; x^8/8! - 5400 \; x^{12}/12! + ... ,$$

so two further empirical checks are whether

$ A060638(8,4) = A214916(12)=88704$ and $A060638(9,4) = A214916(16)=21288960.$

(Cross-posted on MathOverflow.)

1 Answers 1

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$$z= 1-x^4/(3 \cdot 4)+x^8/(3 \cdot 4 \cdot 7 \cdot 8) - x^{12}/(3 \cdot 4 \cdot 7 \cdot 8 \cdot 11 \cdot 12) + .... $$ The calculus below leads to the relation to the Bessel function : $$z=\Gamma\left(\frac{3}{4}\right) \sqrt{\frac{x}{2}}\;J_{-1/4}\left(\frac{x^2}{2}\right)$$

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NOTE :

It is easy to solve $$\frac{d^2 z}{dz^2}=-x^2 z$$ which is an ODE of the Bessel kind. $$z=c_1\sqrt{x}\;J_{1/4}\left(\frac{x^2}{2}\right)+c_2\sqrt{x}\;J_{-1/4}\left(\frac{x^2}{2}\right)$$ This confirmes that the infinite series considered above is a particular solution of the ODE.

OTHER EXAMPLES (with the same method) :

$$1+\frac{x^4}{3 \cdot 4}+\frac{x^8}{3 \cdot 4 \cdot 7 \cdot 8} + \frac{x^{12}}{3 \cdot 4 \cdot 7 \cdot 8 \cdot 11 \cdot 12} + .... = \Gamma\left(\frac{3}{4}\right)\sqrt{\frac{x}{2}}\;I_{-1/4}\left(\frac{x^2}{2}\right)$$ $I_\nu(X)$ is the modified Bessel function of the first kind.

$$1-\frac{x^4}{4 \cdot 5}+\frac{x^8}{4 \cdot 5 \cdot 8 \cdot 9} - \frac{x^{12}}{4 \cdot 5 \cdot 8 \cdot 9 \cdot 12 \cdot 13} + .... = \Gamma\left(\frac{5}{4}\right)\sqrt{\frac{2}{x}}\;J_{1/4}\left(\frac{x^2}{2}\right)$$

$$1+\frac{x^4}{4 \cdot 5}+\frac{x^8}{4 \cdot 5 \cdot 8 \cdot 9} + \frac{x^{12}}{4 \cdot 5 \cdot 8 \cdot 9 \cdot 12 \cdot 13} + .... = \Gamma\left(\frac{5}{4}\right)\sqrt{\frac{2}{x}}\;I_{1/4}\left(\frac{x^2}{2}\right)$$

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    Nice. Of course, $z'' = x^2z $ satisfies the recursion relation for the subset of the entry. My primary interest is the potential connection to the zonotopes. – 2017-02-01
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    Yes, I understand what is tour interest. My concern was not so large, just showing how one can relate a given series with denominators (1,12,672,88704...) to a special function (Bessel in this case), thanks to the properties of some hypergeometric function. This can be done for many other series thanks to the same method, without using an ODE. . – 2017-02-01
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    A general soln to the given Riccati eqn is presented in Project 8.6 Riccati Equation and Modified Bessel Functions http://wps.prenhall.com/esm_edwards_bvp_4/71/18237/4668853.cw/index.html – 2017-02-12
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    $$y(x) = \frac{J_{3/4}(x^2/2)-cJ_{-3/4}(x^2/2)}{cJ_{1/4}(x^2/2)+J_{-1/4}(x^2/2)}$$ – 2017-02-12